Internal problem ID [8241]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 661.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_symmetry_[F(x),G(x)]]]
Solve \begin {gather*} \boxed {y^{\prime }+\frac {a x}{2}+\frac {b}{2}-x^{2} \sqrt {a^{2} x^{2}+2 a b x +b^{2}+4 a y-4 c}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.398 (sec). Leaf size: 39
dsolve(diff(y(x),x) = -1/2*a*x-1/2*b+x^2*(a^2*x^2+2*a*b*x+b^2+4*a*y(x)-4*c)^(1/2),y(x), singsol=all)
\[ c_{1}+\frac {2 a \,x^{3}}{3}-\sqrt {a^{2} x^{2}+2 b x a +4 a y \relax (x )+b^{2}-4 c} = 0 \]
✓ Solution by Mathematica
Time used: 0.317 (sec). Leaf size: 55
DSolve[y'[x] == -1/2*b - (a*x)/2 + x^2*Sqrt[b^2 - 4*c + 2*a*b*x + a^2*x^2 + 4*a*y[x]],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {b^2 \log ^2\left (-\frac {e^{-\frac {4 a \left (x^3-3 c_1\right )}{3 b}}}{a}\right )-4 (a x+b)^2+16 c}{16 a} \\ \end{align*}